Aircraft: EMB ERJ 170/175 ER/LR
Phase of flight: Initial Climb
Anomaly: ATC Issue All Types; Conflict Ground Conflict; Less Severe; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter CFTT / CFIT

We were holding short of RWY 16 at XNA. [An aircraft] had just been cleared for take off on a runway heading. Additionally; there was an aircraft inbound about 7 miles out on the ILS 16. We were then cleared for take off via the Highfill 8 turn heading 180.METAR at the time: XNA 08005KT 4SM -RA BR SCT005 OVC040 17/16 A3017 I was the Non Flying Pilot and during the take roll; I noticed radar showed moderate rain (yellow) at the 180 heading. Once airborne and before the 400 feet heading call; I asked the tower for runway heading to avoid the weather. Tower approved the runway heading and at approximately 2;500 feet to 3;000 feet we received a Terrain Terrain alert. I looked first at my Radar Altimeter which was reading and flickering between 0 and - 5; our altitude was now over 3;000 feet; and we were climbing over 1500 FPM. I stated I thought it was a spurious error; but due to the marginal weather conditions the First Officerfollowed his instincts and went to Max Thrust. At about 4;000 feet I pulled up the ILS 34 Jepp chart and noted a Tower 1;855 feet SE of the Localizer and about 8 km SE of the airport. I also noted that our GPS position was showing on this chart and we were on the outbound LOC course. Additionally; there is no Obstacle DP for RWY 16 on the 20-9 page. I still believe this was a spurious Radar Atl error but while troubleshooting I may have distracted the First Officerfrom making the Terrain Max Thrust call. Considering the marginal visual conditions I should have stated Terrain instead of Troubleshooting. We tookoff at XNA on Runway 16. Just after takeoff; as per the departure procedure we were supposed to turn right to 180 heading. Right at about 400 feet; when I'm supposed to call heading; the Captain asked ATC for runway heading to avoid some bad weather. ATC gave us runway heading and the Captain turned the heading bug to runway heading and selected HDG. At about 1;000 feet; I called Vnav. Then at about 3;000 feet; we had a terrain alert; 'Too low terrain.' I was a little startled by the alert; and thought it must have been an erroneous alert and was somewhat confused because of it. I did pitch up a little more due to the alert; but I did not go max thrust and do a proper terrain escape procedure; even though I still should have.After pitching up a little more for a few seconds I noticed the speed start to drop which I stated to the Captain; and thinking that I should have just done the proper procedure; I then went to max thrust. He stated to just lower the nose a little bit; at which point I did; and then pulled the thrust levers back out of max thrust and started to go into the recovery. The speed started to come back up; and then I called for flaps 1; and flaps up. We switched over to the departure frequency; and then I asked him for the auto throttle back on; and the autopilot on. The flight continued normally without any further incidents. Once we were all cleaned up and climbing properly; we looked over the charts; to see what could have caused the terrain alert. It is not a mountainous terrain airport and the elevation all looks constant. There looks to be one obstacle that may have caused it that goes up to 1;855 feet about 8 miles SE of the field. Both the Captain and I seemed to believe that it was a false indication for the alert; because an [aircraft] took off a few minutes before we did; and the tower told them to fly runway heading right after they were airborne; which I doubt tower would have given them if there were obstacles or terrain in the way. Because we thought it was a false indication was why we did not do a proper escape and recovery; which is not a good excuse. Next time I will be sure to do a proper procedure; no matter what I may think or feel. E175 flight crew reported ATC approved weather avoidance clearance where crew received a terrain alert that the Captain doubted resulted in failure to call and allow First Officer proper escape and recover maneuver.